Answer :
To predict the peak income using the quadratic function of best fit, we follow these steps:
1. Understand the Given Quadratic Function:
The quadratic function that models the income in relation to age is given by:
[tex]\[ y = -41.891 x^2 + 3987.648 x - 49377.617 \][/tex]
where [tex]\( y \)[/tex] represents the income and [tex]\( x \)[/tex] represents the age.
2. Determine the Age at which Peak Income Occurs:
From part (c), the age at which an individual can expect to earn the most income is 47.6 years. Therefore, [tex]\( x = 47.6 \)[/tex] is used to find the peak income.
3. Predict the Peak Income:
To find the peak income, substitute [tex]\( x = 47.6 \)[/tex] into the quadratic function:
[tex]\[ y = -41.891 (47.6)^2 + 3987.648 (47.6) - 49377.617 \][/tex]
Based on the given solution, calculating this expression will yield:
[tex]\[ y \approx 45519.475640000004 \][/tex]
Rounding this to the nearest dollar, the predicted peak income is:
[tex]\[ \$45,519 \][/tex]
Thus, the predicted peak income is about \$45,519.
1. Understand the Given Quadratic Function:
The quadratic function that models the income in relation to age is given by:
[tex]\[ y = -41.891 x^2 + 3987.648 x - 49377.617 \][/tex]
where [tex]\( y \)[/tex] represents the income and [tex]\( x \)[/tex] represents the age.
2. Determine the Age at which Peak Income Occurs:
From part (c), the age at which an individual can expect to earn the most income is 47.6 years. Therefore, [tex]\( x = 47.6 \)[/tex] is used to find the peak income.
3. Predict the Peak Income:
To find the peak income, substitute [tex]\( x = 47.6 \)[/tex] into the quadratic function:
[tex]\[ y = -41.891 (47.6)^2 + 3987.648 (47.6) - 49377.617 \][/tex]
Based on the given solution, calculating this expression will yield:
[tex]\[ y \approx 45519.475640000004 \][/tex]
Rounding this to the nearest dollar, the predicted peak income is:
[tex]\[ \$45,519 \][/tex]
Thus, the predicted peak income is about \$45,519.